International Mathematics Competition
for University Students
2023

IMC2023: Day 1, Problem 1

Problem 1. Find all functions \(\displaystyle f:\) \(\displaystyle \mathbb{R}\to \mathbb{R}\) that have a continuous second derivative and for which the equality \(\displaystyle f(7x+1) = 49f(x)\) holds for all \(\displaystyle x\in\mathbb{R}\).

Alex Avdiushenko, Neapolis University Paphos, Cyprus

Hint:

• The fixed point of \(\displaystyle 7x+1\) is \(\displaystyle -1/6\).
• Differentiating twice cancels out the coefficient \(\displaystyle 49\).

  Solution